(a) The point $(−2, k)$ is on the circle $(x − 2)^2 + (y − 3)^2 = 65$ - Leaving Cert Mathematics - Question 3 - 2019

The center of the circle $s$ will be at $(g, g)$, where $g$ is the radius of the circle since it touches both axes.
The distance from the center $(g, g)$ to the line $3x - 4y + 6 = 0$ should equal the radius.

First, we compute the perpendicular distance from the center to the line using:

$d = \frac{|3g - 4g + 6|}{\sqrt{3^2 + (-4)^2}} = \frac{|−g + 6|}{5}$

Setting this equal to the radius gives:

$|−g + 6| = g$

This simplifies to two equations:

1. $−g + 6 = g$, which gives $g = 3$.
2. $−g + 6 = −g$, which does not provide a valid solution.

Thus, the center is $(3, 3)$ and the radius is $3$. The equation of the circle can be expressed as:

$(x - 3)^2 + (y - 3)^2 = 3^2$

This can be expanded to give:

$x^2 + y^2 - 6x - 6y + 18 = 0$